Suppose we have a mass m moving horizontally with velocity v on a frictionless surface. Directly in its path is a mass M, backed by a spring at resting length with spring constant k.
Now, what will be the maximum distance by which the spring is compressed if the collision between the masses is:
I) perfectly inelastic?
II) perfectly elastic?
I) Inelastic collision means that the masses “stick” when they collide. We use conservation of momentum: the initial momentum is mv, and so for post-collision velocity vf,
Now, to find the maximum compression of the spring, the simplest method is to use energy conservation: for displacement x from resting length, the potential energy in the spring is . Immediately after the collision, we have only the kinetic energy of the joined masses, so the energy is
Maximum compression of the spring occurs when the masses have zero velocity; the kinetic energy is zero, and so the energy is entirely potential, giving us:
II) A perfectly elastic collision conserves energy as well as momentum. Let vf be the post-collision velocity of mass m, and vF that of mass M. Conservation of momentum gives:
and conservation of energy gives:
Solving the first equation for vf,
Substituting into the energy result:
As vF≠0, we have
(which is double the velocity of the inelastic case).
Note that we do not need vf, as now only mass M is compressing the spring.
We have energy
and so at maximum compression,